One application of the combinatorial Laplacian is the celebrated matrix-tree theorem. We state a version on principal minors of the combinatorial Laplacian for edge-weighted graphs, which may be deduced from the all minors matrix-tree theorem in [1]}. We will see that each principal minor of \(L_{(G,\varepsilon )}\)  has a combinatorial interpretation in terms of rooted spanning forests, which consist of a spanning forest of \(G\) , together with a choice of a vertex from each connected component of the spanning forest, constituting the roots. A spanning forest is \(k\) -rooted if it has \(k\)  roots, and \(S\) -rooted for \(S\subseteq V\)  if its roots are given by \(S\) .
